Gaussian Elimination This worksheet is an accompaniment to Section 1.2, Lay's Linear Algebra Written by Michael K. May, S.J., revised by Russell Blyth. Revised by Harry S. Mills restart: with(LinearAlgebra): with(plots): with(plottools):

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Outline: This document is organized by section as follows: Part 1 Express the general solution of a linear system in parametric form. Part 2 Exercises Part 3 Converting between systems of equations and matrices GenerateMatrix for a list of equations and a list of variables. GenerateEquations for a matrix and a list of variables. Part 4 Exercises Part 5 Using Maple commands for elementary row operations RowOperation - Has 3 variations which cover all 3 elementary row operations RandomMatrix - Creates a matrix with random entries. We won't use this a lot right away, but it's something handy that we will use on occasion throughout the semester. Part 6 Using more general (powerful) Maple commands on matrices. ReducedRowEchelonForm - Computes the reduced (row) ecelon form GaussianElimination - Computes an echelon form. Rank - Returns the rank of a matrix (preview) Transpose - Returns the transpose of a matrix. LinearSolve - Solves a system of linear equations - NOTE: We use the "free = t" optional argument to clean up the output when there is more than one solution. Part 7 Exercise

Consider the system {x+2y+3z=6} of one equation in 3 variables. It is obvious that the solution is simply a plane. If we ask Maple to solve the system we get a solution that defines some of the variables in terms of themselves. eq01 := x+2*y+3*z=6; solve({eq01});

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This indicates that we have a point in the solution for any pair of values assigned to y and z. We say that y and z are free variables, while x is a basic (pivot) variable. In terms of college algebra, we can only solve for one variable, in which case we might as well solve for x. solve(eq01,{x});

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We would like to produce the translation vector, and spanning vectors for the solution in parametric form. The parametric form of the solution is a vector written in terms of the free variables. For this system it is: [-2y-3z+6, y, z], which in Maple, is given in vector form by: 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

The translation vector is found by setting the free variables to constant values and simplifying. (Thus one can have many correct translation vectors.) Usually we use the simplest values: set the free variables to 0. This gives a translation vector of [6, 0, 0] for our system. The spanning vectors are obtained by taking the vectors corresponding to the coefficients of the free variables. For this system a set of spanning vectors is {[-2, 1, 0], [-3, 0, 1]}. In Part 6, below, we will look at a way of automating the process of writing a solution to a system in parametric form. This will be more a preview of coming attractions in Section 1.5 in the text, rather than something you must know right now for Section 1.2.

2a. Find the general solution to the system {x-y+2*z-2*w=1, 2*x+y+3*w=4}. Express your answer in parametric form. The solve command will not necessarily follow our conventions for deciding which variables will be free and which will be basic. The choice is arbitrary, as a matter of fact, but in the future, we will want, for example, x in terms of z and w, instead of w in terms of x and z. Our basic variables will correspond to leading entries in the echelon forms. Use the naming convention eqn2a1, eqn2a2... Recall: We used the solve command in the first worksheet to solve a system of 2 equations in 2 variables. Refer back to 01 - VisualSystems, Part 1, for a refresher. 2b. Find the general solution to the system {4x -2y -z -w =1, x +3y -2z -2w =2}. Express your answer in parametric form. Use the convention eqn2b1, eqn2b2

The text points out that the Gaussian elimination method we used on systems of equations can be performed in shorthand by working on the matrix of coefficients of the system. In Maple the command for converting from a system of equations to a matrix is GenerateMatrix. To convert back from a matrix to a system of equations, the command is GenerateEquations. Each of these commands has two forms, one for the matrix of coefficients, and one for the augmented matrix. These commands are part of the LinearAlgebra package. To convert from a system of equations to a matrix we first need a list of equations, and an ordered list of variables. eq031 := x + y + 2*z + w = 1; eq032 := 3*x - 4*y + z + w = 2; eq033 := 4*x - 3*y + 3*z + 2*w = 3; eqlist := [eq031, eq032, eq033]; #create and name the list of eqns. varlist := [x, y, z, w]; #create and name the list of vars.

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If we want the matrix of coefficients, the command GenerateMatrix has three parameters: the list of equations, the list of variables, and the name `augmented`. It produces the augmented matrix. M1 := GenerateMatrix(eqlist, varlist, augmented);

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

If we leave out the `augmented` option, the command returns an ordered pair composed of the coefficient matrix and the vector of constants. This can be handy if we want to work with the coefficient matrix and the right-hand side (vector) separately (preview of matrix-vector equations and products, which we will see in Sections 1.3 and 1.4). (M2,b) := GenerateMatrix(eqlist, varlist);

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

The matrix M2 is the coefficient matrix: M2;

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

The vector b is the vector (3 by 1 matrix) whose entries comprise the right-hand side of the equations in the system: b;

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Yet another way to obtain the coefficient matrix is by referring Maple to the 1st entry in the output for (3.3): M2 := GenerateMatrix(eqlist, varlist)[1];

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

The right-hand side can be plucked out as a separate vector (m by 1 matrix) in a similar way (as a vector), by referring to the 2nd part of the output in (3.3): b:=GenerateMatrix(eqlist,varlist)[2];

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To convert from a matrix to a system of equations we use the GenerateEquations command. To produce a system with the constants set to zero (that is, a homogeneous system) we use two parameters: the matrix of coefficients and the variable list. For a nonhomogeneous system, we either use a third parameter, a vector of constants, or we use the augmented matrix. Recall, M2 is the coefficient matrix, representing the left-hand side of the system. The following command assumes that the right-hand side of each equation is zero. homsys := GenerateEquations(M2,varlist);

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The following command tells Maple that the vector b is to supply the right-hand side, using M2 as the coefficient matrix of the left-hand side.. nonhomsys2 := GenerateEquations(M2,varlist,b);

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

The above is the original system. Finally, we can re-constitute the original system by de-constructing the augmented matrix M1: nonhomsys := GenerateEquations(M1,varlist);

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

Apparently, Maple remembers that M1 was an augmented matrix. Actually, the fact that the (augmented) matrix M1 has 5 columns, and the list of variables varlist contains only 4 variables is why Maple assumes the 5th column contains the right-hand sides of the respective equation, and that M1 is an augmented matrix of the form [A | b].

4a. Use Maple to convert the system{x-y+2z-2w=1, 2x+y+3w=4, 2x+3y+2z=6} to an augmented matrix. Use the convention eq4a1, eq4a2, etc. 4b. Use Maple to convert the augmented matrix 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 to a system of equations in x, y, z, and w. I suggest using the Matrix palette to the left to make it go quickly... Give the matrix the name M4b: 4c. Use the command M4c := RandomMatrix(3, 5); to generate a random 3 by 5 matrix named M4c. Convert it to a system of equations in x, y, z, and w.

Once we have a system of equations converted to an augmented matrix, the next task is to use elementary row operations to perform Gaussian elimination on the matrix. The linalg package of Maple has a command corresponding to each type of elementary row operation. The command RowOperation(M, [r1, r2], scal); is used to add scal times row r2 of matrix M to row r1. The command RowOperation(M, r1, scal); is used to multiply row r1 of M by scal. The command RowOperation(M, [r1, r2]); is used to switch (interchange) rows r1 and r2 of the matrix M. We use these operations to reduce the matrix M1 (from Part 3) above to row echelon form. print(M1);

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

Replace Row 2 by -3*Row 1 + Row 2: M5a := RowOperation(M1,[2,1],-3);

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Replace Row 3 by -4 Row 1 + Row 3: M5b := RowOperation(M5a,[3,1],-4);

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